House leaders worked Monday afternoon to firm up the votes they would need to override Governor Deval Patrick’s threatened veto of a major transportation and taxes bill that the governor believes does not go far enough to increase taxes and spending.

House officials backed off earlier predictions that they expected to narrowly exceed the necessary number of votes needed to turn back the governor. A one- or two-vote margin that officials cited earlier Monday was, by mid-afternoon, more fluid, the officials said.

House and Senate leaders have pushed a $500 million tax package to bankroll transportation projects, in response to Patrick’s $1.9 billion proposal. With a House vote expected today on the bill, Patrick over the weekend dialed up the pressure on the Legislature to agree to his tax plan, with administration officials calling state legislators to press a bully-pulpit advantage in the governor’s fight with legislative leaders.

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Patrick is pushing for enough votes to sustain his threatened veto of legislative leaders’ proposed transportation bill. Officials familiar with the conversations said Patrick aides used unusually transactional language, reminding lawmakers that Patrick wields influence over whether transportation projects are built in their districts. Patrick also has control of the state Democratic Party, whose political operation could make the electoral case to the rank and file to break with legislative leaders.

House budget chairman Brian Dempsey on Friday told reporters that House leaders did not intend to cede ground to Patrick.

Patrick has also engaged outside groups to lobby lawmakers for a higher bottom line, which he calls necessary to meet the state’s infrastructure and economic needs. At least four people were arrested outside the State House on Monday while protesting fare increases for the paratransit service known as The Ride

House Speaker Robert DeLeo has countered that, in the current economic environment, the state’s middle class cannot afford the higher income taxes prescribed by Patrick.

Senate President Therese Murray has lined up with DeLeo, leading to a 2-on-1 dynamic that has helped escalate tensions on Beacon Hill to an unusual pitch. The Senate plans to vote on the House-Senate compromise on Thursday.